Skip to content Skip to navigation Skip to collection information

OpenStax_CNX

You are here: Home » Content » Digital Signal Processing » Fourier Transforms as Unitary Operators

Navigation

Recently Viewed

This feature requires Javascript to be enabled.
 

Fourier Transforms as Unitary Operators

Module by: Mark A. Davenport. E-mail the author

Fourier transforms as unitary operators

We have just seen that the DTFT can be viewed as a unitary operator between 2(Z)2(Z) and L2[-π,π]L2[-π,π]. One can repeat this process for each Fourier transform pair. In fact due to the symmetry between the DTFT and the CTFS, we have already established this for CTFS, i.e.,

CTFS: L 2 [ - π , π ] 2 ( Z ) CTFS: L 2 [ - π , π ] 2 ( Z )
(1)

is a unitary operator. Similarly, we have

CTFS: L 2 ( R ) L 2 ( R ) CTFS: L 2 ( R ) L 2 ( R )
(2)

is a unitary operator as well. The proof of this fact closely mirrors the proof for the DTFT. Finally, we also have

DFT: C N C N . DFT: C N C N .
(3)

This operator is also unitary, which can be easily verified by showing that the DFT matrix is actually a unitary matrix: UHU=UUH=IUHU=UUH=I.

Note that this discussion only applies to finite-energy (2/L22/L2) signals. Whenever we talk about infinite-energy functions (things like the unit step, delta functions, the all-constant signal) having a Fourier transform, we need to be very careful about whether we are talking about a truly convergent Fourier representation or whether we are merely using an engineering “trick” or convention.

Collection Navigation

Content actions

Download:

Collection as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add:

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks